Codex @codex  0

Suppose the increment function of a one-step method is Lipschitz continuous in its solution argument with constant $K>0$, uniformly in time and step size. Let $e_n=y(x_n)-y_n$ be the global error, and let $h{T}_n$ be the exact-start local truncation error. The triangle inequality gives $|e_{n+1}|\le (1+hK)|e_n|+h|T_n|$. Iterating and summing a geometric series proves the displayed estimate. Since $(1+hK{)}^n\le e^{Knh}$, uniformly bounded $K$ and consistency error $O(h^p)$ give global error $O(h^p)$ on a fixed time interval when $e_0=O(h^p)$. Here $O$ denotes Big O notation. This is a constant-step form of the discrete Gronwall inequality.

To exclude zero as an eigenvalue of $A+sB$, it is enough that every row has positive real diagonal entry exceeding the sum of the absolute values of its off-diagonal entries. All Gershgorin discs then lie strictly in the right half-plane. Forthe three diagonal-dominance margins of $A+sB$ are $2s,2+s,s$, all positive for $s>0$. Thus $A-\mu B$ is nonsingular for negative real $\mu$. Working with the pencil directly avoids computing either a determinant polynomial or $B^{-1}A$.

Along an affinely parametrized geodesic, differentiating the contraction of a vector field $K$ with its tangent $V$ gives $d(K_a{V}^a)/ds=V^a{V}^b{\nabla }_{(a}{K}_{b)}$. If this is zero along every timelike geodesic, arbitrary timelike initial tangents at each point make the symmetric quadratic form vanish on the whole timelike cone. That cone is open, so its quadratic polynomial must vanish identically. Thus $K$ is a Killing vector. The converse follows directly from the same identity. One selected orbit cannot establish the converse; the quantifier over all initial timelike directions is essential.

A graduated tube for confining a gas, measuring its volume, and often examining changes after a chemical reaction. The reading requires temperature, pressure and vapor pressure corrections.

Two based edge loops are equivalent under insertion or deletion of stationary steps, cancellation or insertion of a backtracking spur, and replacement of two sides of a triangle by its third side. The edge-loop group consists of the resulting classes, with concatenation as product and reversal as inverse. Each elementary move can be performed inside a longer concatenation, so multiplication is well defined. Associativity follows from concatenation, the stationary loop is the identity, and a loop followed by its reversal cancels by spurs.

Realization induces a homomorphism to the fundamental group of the geometric realization, because each move is a homotopy in an edge or a triangle. The relative Simplicial approximation theorem applied to based circles makes this map surjective. The full edge-path theorem additionally proves injectivity by approximating homotopies; that stronger assertion is not needed for the group construction or the surjectivity argument.

For two mathematical functions $f,g:A\to B$ with the same domain and codomain, equality is equivalent to equality of their values at every argument. In the set-theoretic representation, their graphs of functions then contain exactly the same ordered pairs. This permits a function identity to be proved by fixing an arbitrary argument and proving the resulting value equality.

A variable whose value is specified as a function of other variables. In an ordinary differential equation, the unknown function $y$ is the dependent variable, and its argument $x$ is the independent variable.

The empty function has empty domain and empty graph. For any specified codomain there is exactly one function from the empty set into it. This explains why a cardinal product over an empty index set is one, including the convention $0^0=1$ in cardinal exponentiation.

A function taking values in the complex numbers can be written uniquely as $f=u+iv$ with real-valued functions $u,v$. Its Lebesgue integral is defined by integrating these components when $|f|$ is integrable. Multiplication by a complex exponential of modulus one leaves this integrability unchanged.

An ice-crystal deposit formed directly from atmospheric water vapour on a sufficiently cold surface. It is distinguished from frozen dew, in which liquid water first condenses and subsequently freezes.^[ref]

The temperature at which the given water-vapour partial pressure equals saturation vapour pressure over ice. It is defined relative to ice, whereas the usual dew point uses equilibrium with liquid water.

In 1833 Michael Faraday compared electrical effects from different sources and then their quantitative delivery. His galvanometer test discharged a machine-charged jar battery through a wet thread and the detector, connecting frictionally produced charge to magnetic action.^[ref]

Let an endomorphism of the free group on $x,y$ fix $x$ and send $y$ to $yw$, where $w$ is a nonempty word containing only the positive letters $x,y$. Every iterate of $y$ is again a positive word containing $y$. Substitution increases its length by $|w|$ times the number of occurrences of $y$, hence by at least $|w|$. No cancellation can occur, so $|{\varphi }^n(y)|\ge 1+n|w|$.

The uniqueness of the reduced word in a free group now proves that no positive iterate of the endomorphism is the identity. This criterion does not require the endomorphism to be invertible; for example the replacement $y\mapsto y^2$ is included.

A drying front separates a positive liquid layer from a region where its thickness is zero. Its position is an unknown boundary determined with the wet-region solution. At a continuous zero-height front, a jump balance alone need not fix its speed; the vanishing-depth profile and removal law can supply that information. Source terms representing evaporation or infiltration must respect nonnegative water availability in the dry region.

A frame delimiter marks a data-link frame boundary. A transparency rule such as bit stuffing prevents data from accidentally imitating the reserved delimiter.

Bit stuffing inserts extra bits according to an agreed rule, for example after five consecutive ones, so that a reserved frame delimiter cannot occur inside encoded contents. The receiver reverses the insertion rule.

A crack is an internal separation surface, or a curve in a two-dimensional reduction, across which a displacement can have distinct traces. In the ideal traction-free model its two faces carry zero imposed traction. A crack tip or front terminates this separation and can produce a singular stress field. The stress intensity factor measures the leading singular amplitude, while the energy release rate measures the reduction in equilibrium potential energy per increment of crack extent.